On pro-${\it p}$-groups with a single defining relator

TL;DR

This paper investigates conditions under which the Freiheitsatz of Magnus, originally for free groups, can be extended to free pro-p-groups, providing insights into their algebraic structure.

Contribution

It establishes necessary conditions for extending Magnus's Freiheitsatz to free pro-p-groups, advancing understanding of their defining relations.

Findings

Identifies necessary conditions for Freiheitsatz extension

Provides new criteria for pro-p-group presentations

Enhances theoretical understanding of pro-p-group structure

Abstract

In this paper we establish necessary condition for the Freiheitsatz of Magnus can be extended to free pro-${\it p}$-groups